I don not understand. I assume you want to take Z = {your non-isolated point}, but this will in general not be a connected component of U.

@Marc Hoyois: The example you gave is not a counterexample, since $Z$ is supposed to be closed in $X$. But in your setup the only closed set contained in U is the empty set.

The scheme $X$ may indeed not be Noetherian, thus the connected components are not open in general. I added the qcqs condition so that quasi-components and connected components are the same, but I do not know whether this condition is needed.